The total income of $A$ and $B$ is greater than the total income of $C$ and $D$. The total income of $A$ and $C$ is equal to the total income of $B$ and $D$. Furthermore,the income of $A$ is equal to half the total income of $B$ and $D$. Which of the following statements is $NOT$ true?

$(i)$ Five friends $P, Q, R, S$ and $T$ travelled to five different cities of Chennai,Calcutta,Delhi,Bangalore and Hyderabad by five different modes of transport of Bus,Train,Aeroplane,Car and Boat from Mumbai.
$(ii)$ The person who travelled to Delhi did not travel by boat.
$(iii)$ $R$ went to Bangalore by car and $Q$ went to Calcutta by aeroplane.
$(iv)$ $S$ travelled by boat whereas $T$ travelled by train.
$(v)$ Mumbai is not connected by bus to Delhi and Chennai.
Which of the following combinations of person and mode is not correct?

In a pile of $10$ books,there are $3$ History,$3$ Hindi,$2$ Mathematics,and $2$ English books. From the top,an English book is between a History and a Mathematics book,a History book is between a Mathematics and an English book,a Hindi book is between an English and a Mathematics book,a Mathematics book is between two Hindi books,and two Hindi books are between a Mathematics and a History book. Which subject book is at the $6^{th}$ position from the top?

There are five persons $P, Q, R, S$ and $T$. One is a football player,one is a chess player,and one is a hockey player. $P$ and $S$ are unmarried ladies and do not participate in any game. None of the ladies plays chess or football. There is a married couple in which $T$ is the husband. $Q$ is the brother of $R$ and is neither a chess player nor a hockey player. Who is the hockey player?

$A, B, C, D,$ and $E$ are five persons standing in a bus queue. $A$ is a teacher standing to the right of $D$ who is an advocate. $B$ is a scientist and is standing to the left of $E$ who is an electrician. There are two persons between $B$ and $A$. $C$ who is a banker stands to the right of $A$. Who is to the extreme left?

Prashant Arora has three children $-$ Sangeeta,Vimal,and Ashish. Ashish married Monika,the eldest daughter of Mr. and Mrs. Roy. The Roys married their youngest daughter to the eldest son of Mr. and Mrs. Sharma,and they had two children named Amit and Shashi. The Roys have two more children,Roshan and Vandana,both elder to Veena. Sameer and Ajay are sons of Ashish and Monika. Rashmi is the daughter of Amit. What is the surname of Rashmi?